Representation of magnetic fields in space

Stern, David P.

doi:10.1029/rg014i002p00199

Cited by 115 publications

(79 citation statements)

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“…However, the tracing procedure can be greatly simplified, at least for an axisymmetric B field derived from a scalar potential, by formally constructing Euler potentials ; such that and B r 2 r ÿ r V (e.g., Stern, 1976Stern, , 1994 explicitly as a factor common to all terms in the summation, leaving the derivative (rather than the integral) of P n x with respect to x to be generated (as a callable function) by Mathematica (Wolfram, 1992).…”

Section: Test Casementioning

confidence: 99%

Non-spherical source-surface model of the heliosphere: a scalar formulation

Schulz

1997

Ann. Geophys.

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Abstract. The source-surface method offers an alternative to full MHD simulation of the heliosphere. It entails specification of a surface from which the solar wind flows normally outward along straight lines. Compatibility with MHD results requires this (source) surface to be non-spherical in general and prolate (aligned with the solar dipole axis) in prototypical axisymmetric cases. Mid-latitude features on the source surface thus map to significantly lower latitudes in the heliosphere. The model is usually implemented by deriving the B field (in the region surrounded by the source surface) from a scalar potential formally expanded in spherical harmonics, with coefficients chosen so as to minimize the mean-square tangential component of B over this surface. In the simplified (scalar) version the quantity minimized is instead the variance of the scalar potential over the source surface. The scalar formulation greatly reduces the time required to compute required matrix elements, while imposing essentially the same physical boundary condition as the vector formulation (viz., that the coronal magnetic field be, as nearly as possible, normal to the source surface for continuity with the heliosphere). The source surface proposed for actual application is a surface of constantF r ÿ kB , where r is the heliocentric distance andB is the scalar magnitude of the B field produced by currents inside the Sun. Comparison with MHD simulations suggests that k 1:4 is a good choice for the adjustable exponent. This value has been shown to map the neutral line on the source surface during Carrington Rotation 1869 (May-June 1993) to a range of latitudes that would have just grazed the position of Ulysses during that month in which sector structure disappeared from Ulysses' magnetometer observations. BackgroundThe magnetic field in the solar corona is traditionally modeled (Schatten et al., 1969;Altschuler and Newkirk, 1969;Hoeksema and Scherrer, 1986;Hoeksema, 1991) as being current-free (and thus derivable from a scalar potential) within a spherically annular volume of inner radius 1 r in order to achieve better agreement with dimensions of large coronal helmet structures seen in eclipse photographs.Subsequent magnetohydrodynamic (MHD) simulations by Pneuman and Kopp (1971a, b) showed (as many eclipse photographs had already indicated) that the outflow of solar wind must be significantly nonradial in the inner heliosphere. The dashed curves in .

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Section: Test Casementioning

confidence: 99%

Non-spherical source-surface model of the heliosphere: a scalar formulation

Schulz

1997

Ann. Geophys.

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“…Thus, the constraint β = β(µ, Φ) makes the two terms in the current density uncoupled (separately divergence-free). An unmatched Euler potential description can be converted to a regular Euler potential description by an appropriate transform of variables (Stern 1976). Particularly, if β is a function of µ only, we can write…”

Section: Imposition Of Special Constraintsmentioning

confidence: 99%

Magnetic Flux-Current Surfaces of Magnetohydrostatic Equilibria

Choe

Jang²

2013

Journal of The Korean Astronomical Society

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Magnetohydrostatic equilibria, in which the Lorentz force, the plasma pressure force and the gravitational force balance out to zero, are widely adopted as the zeroth order states of many astrophysical plasma structures. A magnetic flux-current surface is a surface, in which both magnetic field lines and current lines lie. We for the first time derive the necessary and sufficient condition for existence of magnetic flux-current surfaces in magnetohydrostatic equilibria. It is also shown that the existence of flux-current surfaces is a necessary (but not sufficient) condition for the ratio of gravity-aligned components of current density and magnetic field to be constant along each field line. However, its necessary and sufficient condition is found to be very restrictive. This finding gives a significant constraint in modeling solar coronal magnetic fields as force-free fields using photospheric magnetic field observations.

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“…; of the Earth's main magnetic field, which is of internal origin, can be expressed in the form (Chapman and Bartels, 1940;Roederer, 1972;Stern, 1976Stern, , 1994Langel, 1992 …”

Section: Spherical Harmonic Analysis Of the Main Fieldmentioning

confidence: 99%

“…More generally, the Earth's magnetic field provides a basic coordinate system for studying the distribution and movement of plasmas and energetic charged particles in the ionosphere and magnetosphere (Chapman and Bartels, 1940;McIlwain, 1961;Akasofu and Chapman, 1972;Stern, 1976Stern, , 1994Stern and Tsyganenko, 1992). For example, precise knowledge of the geomagnetic field is important in detailed studies of: (i) the motion of trapped particles that form the ''Van Allen radiation belts'' (Roederer, 1972;Walt, 1994); (ii) the precipitation of auroral particles into the upper atmosphere (McIlwain, 1960;Albert, 1967;Evans, 1968;Eather, 1973;Meng, 1978;Feldstein and Galperin, 1985;Gorney, 1987;Newell et al, 1991); and (iii) the trajectories of energetic solar protons and galactic cosmic rays in the vicinity of the Earth (St ormer, 1955;Vallarta, 1961;Alfv en and F althammar, 1963;Northrop, 1963;Roederer, 1970;Baker et al, 1990;Shea and Smart, 1990).…”

Section: Introductionmentioning

confidence: 99%

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

1997

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Abstract.The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a ''magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-fieldline tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. g 0 n 6 g 0 n ). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as ''first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

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Representation of magnetic fields in space

Cited by 115 publications

References 100 publications

Non-spherical source-surface model of the heliosphere: a scalar formulation

Non-spherical source-surface model of the heliosphere: a scalar formulation

Magnetic Flux-Current Surfaces of Magnetohydrostatic Equilibria

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

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